Empirical Methods of Economics
1) In the Cobb-Douglas production function, the relation between output, Q, labour input, L, and capital, K, is specified as follows:
Q = áL â1Kâ2
a) Specify the econometric model. Explain how you arrive at the model you specify.
This model is purely deterministic. We do not expect real economies subject to myriad perturbations (or even just rounding errors in the reported data) to strictly adhere to this model, so we need an error term, m, E(m)=0. We might try Q = áL â1Kâ2+m, but then Q(L = 0) = Q(K = 0) =µ, implying that output could be negative (Q < 0) under realistic inputs (L = 0 or K = 0). Because output cannot be negative, we use:
b) Rewrite the model in (a) in a form suitable for estimation by OLS.
c) Assuming constant returns to scale obtain the labour intensive and capital intensive form of the model in (b)
Constant returns to scale is a condition under which scaling the inputs by a factor (l) scales the outputs by the same factor. Using the OLS form of the equation and imposing this condition, we obtain an equivalent result to that of the textbook (Gujarati, 2003):
d) Using the data in Table 1, estimate the models in (b) and (c) using Eviews.
e) Report the results of your estimations.
f) Using t- and F tests what can you conclude from the estimated model.
The F-test tells us that both models fit the data better than would be expected by chance (P < 1e-13). The high R2 for a large number of data points (relative to the degrees of freedom) also suggested this result.
Next we use the R2 of each model, along with 8.7.10 from our book to assess the probability that the restricted model is just as good as the unrestricted model on this dataset (Gujarati, 2003), at a significance level of P = 0.05, (for which the critical value will be F = 4.3, since df in the numerator is 1, and the denominator is 22). In the context of 8.7.10, m = 1 (we eliminated one parameter), n = 24, k = 2, so F = (0.957-0.937)/1 / ((1-0.957)/(24-2) = 0.02 / (0.043/22) = 10.23. Thus we reject the null hypothesis (P < 0.05); the unrestricted model performs better than the restricted model on this data set (the actual probability that the unrestricted model is not statistically better is around P=0.004, using the web-calculator at http://www.danielsoper.com/statcalc/calc04.aspx). In this case it appears that the assumption that there are constant returns to scale is violated, although for many applications on similar data sets, the constant returns model would be adequate, since R2=0.937. The F-test merely tells us that removing the assumption about constant returns is very likely a better improvement than adding another random variable to the model.
Lastly, we can use the t-tests to investigate whether each parameter in each model is significantly different from 0. The intercept term is insignificant in both models (P > 0.05), thus we could set it to 0 (á = 1), without significant effect on the regression. The other parameters are significantly different than 0 (P < 0.05).
Bibliography:
GUJARATI, D. N. (2003) Basic econometrics, Boston, McGraw Hill.
